Lecture notes - pink button - My lecture notes from class.
Suggested homework exercises - As listed.
Section 3.1: Introduction to vectors. Vectors in 2-space and 3-space: geometric and analytic. Equivalent vectors, sum and difference of vectors and the scalar multiple of a vector. Zero vector, negative of a vector, components of a vector, coordinate systems and translation of axes and translation equations..
Pages 140 - 142: 1 -14, 16, 19-21, 25, 31-33 (for R3)
Section 3.2: Properties of vector arithmetic and norm: geometric and analytic. Length (Norm) of a vector, unit vector, distance between two points in 2-space and 3-space.
Pages 153-155: 1-4, 11, 12, 33 (on R3)
Section 3.3. Dot product (Euclidean inner product) and projection of vectors. Finding the angle between two vectors, orthogonal vectors. Properties of the dot product. Orthogonal projection of a vector. The length of the projection vector. Distance between a point and a line.
Pages 140-142 (Dot Product): 9-18, 34, 35
Pages 162-163 (Orthogonality): 1, 2, 29, 30, 31, 32, 33
(Projection): 13, 14, 15-20
(Distance between a point and line): 21-24
Section 3.5: Cross product of vectors. Relating cross product and dot product. Properties of cross product. Standard unit vectors. Writing the cross product in determinant form. Area of a parallelogram and triangle in 3-space. Scalar triple product of vectors in 3-space. Area of a parallelogram in 2-space and the volume of a parallelepiped in 3-space. Determining if vectors in 3-space line in the same plane.
Pages 179-181: 1 - 30, 33, 34, 36, 42
Lines and planes in 3-space.
The normal vector to the plane. Point-normal form, general form and vector form of the equation of a plane. Solution possibilities for planes. Parametric equations for a line and vector form of the equation of a line in 3-space. Distance between a point and a plane.
Pages 162-163(planes): 3-6, 7-10, 11,12, 25, 26, 27, 28
Pages 170-172(lines and planes): 1-14
Euclidean n-space.
Ordered n-tuple and n-space. Vectors in n-space (equality, sum, difference, scalar multiple, zero vector, additive inverse, orthogonal). Properties of vectors in n-space. Euclidean inner product and properties. Euclidean norm and distance and properties. Cauchy-Schwarz inequality in n-space. Relating norm and inner product. Pythagorean theorem in n-space. A matrix formula for dot product. Dot product in view of matrix multiplication.
Pages 140-142: 11, 12, 13, 14, 15, 17, 18, 22, 31-33 (for Rn)
Pages 153-154: 5-12, 15, 16, 17, 18, 33, 34, 35
Pages 162-163: 1c,d , 2d, 20
Section 4.2: Linear Transformations from Rn to Rm. Transformations and operators. Functions, image, domain, codomain and range. Linear transformation, standard matrix, multiplication by A. Geometry of linear transformations. Zero transformation and identity operator. Reflection, projection, rotation, dilation and contraction operators. Compositions of linear transformations.
Eigenvalues
&
Eigenvectors
Section 7.1: Eigenvalues & Eigenvectors.
Eigenvector of A, eigenvalue of A, characteristic equation, characteristic polynomial of A. Eigenvalues of
triangular matrices. Eigenvalues of the powers of a matrix. Eigenvalues and invertibility.
3.5
Eigenvalues
&
Eigenvectors
Section 4.3: Properties of linear transformations from Rn to Rm. One to one linear transformations. Inverse of one to one linear transformations. Linearity properties. Standard basis vectors for Rn.
Liliana Menjivar
Instructor